By Terence Tao
This is a component of a two-volume publication on actual research and is meant for senior undergraduate scholars of arithmetic who've already been uncovered to calculus. The emphasis is on rigour and foundations of research. starting with the development of the quantity structures and set thought, the publication discusses the fundamentals of study (limits, sequence, continuity, differentiation, Riemann integration), via to strength sequence, numerous variable calculus and Fourier research, after which eventually the Lebesgue vital. those are virtually completely set within the concrete surroundings of the true line and Euclidean areas, even supposing there's a few fabric on summary metric and topological areas. The e-book additionally has appendices on mathematical good judgment and the decimal approach. the complete textual content (omitting a few much less imperative issues) might be taught in quarters of 25–30 lectures each one. The path fabric is deeply intertwined with the routines, because it is meant that the coed actively examine the fabric (and perform considering and writing conscientiously) via proving numerous of the major leads to the theory.
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Extra resources for Analysis II: Third Edition
7). 11). In fact, this property is often considered the more fundamental notion of compactness than the sequence-based one. 5. 8 has an important corollary: that every nested sequence of non-empty compact sets is still non-empty. 9. Let (X, d) be a metric space, and let K1 , K2 , K3 , . . be a sequence of non-empty compact subsets of X such that K1 ⊃ K2 ⊃ K3 ⊃ . . Then the intersection ∞ n=1 Kn is non-empty. Proof. 6. We close this section by listing some miscellaneous properties of compact sets.
1. 2. 2. 3. Show that every uniformly continuous function is continuous, but give an example that shows that not every continuous function is uniformly continuous. 4. Let (X, dX ), (Y, dY ), (Z, dZ ) be metric spaces, and let f : X → Y and g : Y → Z be two uniformly continuous functions. Show that g ◦ f : X → Z is also uniformly continuous. 36 2. 5. Let (X, dX ) be a metric space, and let f : X → R and g : X → R be uniformly continuous functions. Show that the direct sum f ⊕g : X → R2 deﬁned by f ⊕ g(x) := (f (x), g(x)) is uniformly continuous.
Continuous functions 29 and let x0 ∈ X be a point in X. Then the following three statements are logically equivalent: (a) f is continuous at x0 . (b) Whenever (x(n) )∞ n=1 is a sequence in X which converges to x0 with respect to the metric dX , the sequence (f (x(n) ))∞ n=1 converges to f (x0 ) with respect to the metric dY . (c) For every open set V ⊂ Y that contains f (x0 ), there exists an open set U ⊂ X containing x0 such that f (U ) ⊆ V . Proof. 1. Another important characterization of continuous functions involves open sets.